Berge-Gabai knots and L-space satellite operations

TL;DR

This paper characterizes when satellite knots with Berge-Gabai pattern knots produce L-space knots, showing it depends on the companion being an L-space knot and the pattern's positive twisting.

Contribution

It generalizes previous cable results by establishing conditions for Berge-Gabai pattern satellite knots to be L-space knots.

Findings

$P(K)$ is an L-space knot iff $K$ is an L-space knot and $P$ is sufficiently positively twisted.

Generalizes Hedden and the first author's cable knot results.

Provides a criterion for Berge-Gabai pattern satellite knots to produce L-space knots.

Abstract

Let $P(K)$ be a satellite knot where the pattern, $P$, is a Berge-Gabai knot (i.e., a knot in the solid torus with a non-trivial solid torus Dehn surgery), and the companion, $K$, is a non-trivial knot in $S^3$. We prove that $P(K)$ is an L-space knot if and only if $K$ is an L-space knot and $P$ is sufficiently positively twisted relative to the genus of $K$. This generalizes the result for cables due to Hedden and the first author.